Transformations of Boolean Functions

Boolean functions are characterized by the unique structure of their solution space. Some properties of the solution space, such as the possible existence of a solution, are well sought after but difficult to obtain. To better reason about such properties, we define transformations as functions that change one Boolean function to another while maintaining some properties of the solution space. We explore transformations of Boolean functions, compactly described as Boolean formulas, where the property is to maintain is the number of solutions in the solution spaces. We first discuss general characteristics of such transformations. Next, we reason about the computational complexity of transforming one Boolean formula to another. Finally, we demonstrate the versatility of transformations by extensively discussing transformations of Boolean formulas to "blocks," which are solution spaces in which the set of solutions makes a prefix of the solution space under a lexicographic order of the variables

Random CNF-XOR Formulas

Boolean Satisfiability (SAT) is fundamental in many diverse areas such as artificial intelligence, formal verification, and biology. Recent universal-hashing based approaches to the problems of sampling and counting crucially depend on the runtime performance of specialized SAT solvers on formulas expressed as the conjunction of both k-CNF constraints and variable-width XOR constraints (known as CNF-XOR formulas), but random CNF-XOR formulas are unexplored in prior work. In this work, we present the first study of random CNF-XOR formulas. We prove that a phase-transition in the satisfiability of random CNF-XOR formulas exists for k=2 and (when the number of k-CNF constraints is small) for k>2. We empirically demonstrate that a state-of-the-art SAT solver scales exponentially on random CNF-XOR formulas across many clause densities, peaking around the empirical phase-transition location. Finally, we prove that the solution space of random CNF-XOR formulas 'shatters' at all nonzero XOR-clause densities into well-separated components

ADDMC: Weighted Model Counting with Algebraic Decision Diagrams

We present an algorithm to compute exact literal-weighted model counts of Boolean formulas in Conjunctive Normal Form. Our algorithm employs dynamic programming and uses Algebraic Decision Diagrams as the primary data structure. We implement this technique in ADDMC, a new model counter. We empirically evaluate various heuristics that can be used with ADDMC. We then compare ADDMC to state-of-the-art exact weighted model counters (Cachet, c2d, d4, and miniC2D) on 1914 standard model counting benchmarks and show that ADDMC significantly improves the virtual best solver.Comment: Presented at AAAI 202

Ising Model Partition Function Computation as a Weighted Counting Problem

While the Ising model remains essential to understand physical phenomena, its natural connection to combinatorial reasoning makes it also one of the best models to probe complex systems in science and engineering. We bring a computational lens to the study of Ising models, where our computer-science perspective is two-fold: On the one hand, we consider the computational complexity of the Ising partition-function problem, or #Ising, and relate it to the logic-based counting of constraint-satisfaction problems, or #CSP. We show that known dichotomy results for #CSP give an easy proof of the hardness of #Ising and provide new intuition on where the difficulty of #Ising comes from. On the other hand, we also show that #Ising can be reduced to Weighted Model Counting (WMC). This enables us to take off-the-shelf model counters and apply them to #Ising. We show that this WMC approach outperforms state-of-the-art specialized tools for #Ising, thereby expanding the range of solvable problems in computational physics.Comment: 16 pages, 2 figure

On Uniformly Sampling Traces of a Transition System (Extended Version)

A key problem in constrained random verification (CRV) concerns generation of input stimuli that result in good coverage of the system's runs in targeted corners of its behavior space. Existing CRV solutions however provide no formal guarantees on the distribution of the system's runs. In this paper, we take a first step towards solving this problem. We present an algorithm based on Algebraic Decision Diagrams for sampling bounded traces (i.e. sequences of states) of a sequential circuit with provable uniformity (or bias) guarantees, while satisfying given constraints. We have implemented our algorithm in a tool called TraceSampler. Extensive experiments show that TraceSampler outperforms alternative approaches that provide similar uniformity guarantees.Comment: Extended version of paper that will appear in proceedings of International Conference on Computer-Aided Design (ICCAD '20); changed wrong text color in sec 7; added 'extended version

Cortico-striatal synaptic defects and OCD-like behaviours in Sapap3-mutant mice

Obsessive-compulsive disorder (OCD) is an anxiety-spectrum disorder characterized by persistent intrusive thoughts (obsessions) and repetitive actions (compulsions). Dysfunction of cortico-striato-thalamo-cortical circuitry is implicated in OCD, though the underlying pathogenic mechanisms are unknown. SAP90/PSD95-associated protein 3 (SAPAP3) is a postsynaptic scaffolding protein at excitatory synapses that is highly expressed in the striatum. Here we show that mice with genetic deletion of SAPAP3 exhibit increased anxiety and compulsive grooming behavior leading to facial hair loss and skin lesions; both behaviors are alleviated by a selective serotonin reuptake inhibitor. Electrophysiological, structural, and biochemical studies of SAPAP3 mutant mice reveal defects in cortico-striatal synapses. Furthermore, lentiviral-mediated selective expression of SAPAP3 in the striatum rescues the synaptic and behavioral defects of SAPAP3 mutant mice. These findings demonstrate a critical role for SAPAP3 at cortico-striatal synapses and emphasize the importance of cortico-striatal circuitry in OCD-like behaviors